Your equation means that nothing can ever be accelerated because the frictional force always opposes the acceleration exactly--but a body in constant motion on a not-frictionless surface will always remain in motion--it can nether accelerate because of friction, nor can it decelerate because friction will work to keep it going.
Clearly something is wrong.
Before working on wheels, consider a block (mass $m$) on a surface:
Here one has to consider the coefficient of static friction, $\mu_s$. Here it requires a force greater than $mg\mu_s$ (the weight of the object times the friction coefficient) to get the motion started. As you ramp your force up from zero, the frictional force matches it and keeps the block motionless. When you pass the threshold, it starts to move.
Once that happens, you consider the coefficient of kinetic friction, $\mu_k$, which is generally less that $\mu_k$. If you apply a horizontal force $F$ to accelerate the object, some portion goes to overcoming friction:
$$ F - mg\mu_k = ma $$.
When considering wheels, the instantaneous point of contact is static--so you use the coefficient of static friction. Under braking circumstances, this is higher than the coefficient of kinetic friction, and this is why we have anti-lock brakes. If the wheels lock up, you're now using $\mu_k$, which provided less stoping power, thereby increasing your stopping distance.
Likewise when accelerating: funny cars than spin their tires lose the race.
Note the frictional force involves a total downward force term $mg$--this is why F1 cars (for example) have wings. The wings are mounted upside down (relative to an airplane), so the lift actually pushes the car down.
This is called "downforce", and is added to the weight of the car to increase the frictional force. Hence, F1 cars can pull 5+ g's in a turn--with the odd effect (for recreational drivers like us) that the faster you go, the more g's you can pull.